It is a standard case, that if we seek a solution to Problem REF  in \(C^1_V\)  that there might not be a solution.
Essentially, it might be too much to ask for existence of a strong material derivative.
For this reason we define the weak derivative as done in [1]}.
The main idea of the weak derivative is to use the transport theorem [2]}, which describes how the inner product on \(H(t)\)  varies in time.
The following version of the Transport theorem appears as Theorem 2.40 of [1]}
